Questions tagged [eigenvalue]

A linear operator (including a matrix) acting on a non-zero *eigenvector* preserves its direction but, in general, scales its magnitude by a scalar quantity *λ* called the *eigenvalue* or characteristic value associated with that eigenvector. Even though it is normally used for linear operators, it may also extend to nonlinear operations, such as Schroeder functional composition, which evoke linear operations.

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419 views

Lower bounds on spectral gaps of ferromagnetic spin-1/2 XXX Hamiltonians?

Question. Are there any references or techniques which can be applied to obtain energy gaps for ferromagnetic XXX spin-1/2 Hamitlonians, on general interaction graphs, or tree-graphs? I'm interested ...
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102 views

Does the trace distance specify a unique state

In quantum information, we frequently use the trace distance (see definition) to look at how similar two states are. If I had a known complete set of states $\{\rho_i\}$ and some unknown state $\...
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Physical meaning of Laplace-Beltrami eigenfunctions?

The eigenfunctions of Laplace-Beltrami operator are often used as the basis of functions defined on some manifolds. It seems that there is some kind of connection between eigen analysis of Laplace-...
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1answer
669 views

What is the problem of having an inertia tensor not satisfying the triangle inequality?

It is well known that rigid body inertia tensors are 3 by 3 positive semidefinite matrices, which is the same as saying that their eigenvalues are all non-negative. A little less known is the fact ...
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162 views

What's the space of eigenvalues/field configurations for a fermion?

In the Schrödinger picture of quantum field theory, the field eigenstates of a real scalar field $\hat\phi(\mathbf x)$ with $\mathbf x \in\mathbb R^3$ are the states $\hat\phi(\mathbf x)|\phi\rangle=\...
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52 views

Chua's Circuit: an inequality ensuring that the equilibrium is not stable

According to Kennedy's Robust op-amp realization of Chua's circuit(1992), the differential equations satisfied by several physical quantities in Chua's circuit are $$\begin{aligned} C_{1} \frac{d v_{...
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216 views

Why is it that a coupled mass-spring system will always produce a diagonalizable matrix?

If you take a system like the one in the image, and you do the $y=x'$ trick to turn it into a first-order system of equations ($x_{1}$ or $x_{2}$ being the displacement of the mass $m_{1}$ or $m_{2}$ ...
3
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219 views

Eigenvalue problem $−\psi''(x) − (ix)^ N \psi(x) = E\psi(x)$ in complex plane

To find the eigenvalue in the complex plane of $x$ for one dimensional Schrodinger equation $$ −ψ''(x) − (ix)^ N ψ(x) = Eψ(x). $$ where $N$ can be any real number, the boundary condition $ψ(x) → 0$ ...
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382 views

Determine the number of bound states admitted in Schrodinger system

Is there a general method for determining the number of bound states admitted by a potential in the Schrodinger equation? Certainly the number of dimensions must factor in somehow: the delta ...
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180 views

Can you write the Einstein equation as an eigenvalue equation for analytical solutions (to local problems) on non-flat spacetimes

In reading about various local solutions to Einstein's field equation it is easy to forget that they almost all assume a flat background spacetime (at least asymptotically). Considering this made me ...
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1answer
87 views

Boundary conditions for the numerical particle in a box example

I want to solve the one-dimensional Schrödinger equation for the particle in a box example, and want to force the wavefunctions to zero on the boundaries. I am using the matrix, \begin{equation} \hat{...
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58 views

Uncertainty Propagation in Coupled Oscillators

I am a senior physics and mathematics major, and this is my last semester. As a result, I am taking advanced physics lab. One of the labs deals with the modal analysis of three spring-mass systems ...
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422 views

Tight-binding model of diatomic solid

I am working on a problem based on Problem 1.6 in Gerald D. Mahan - Many Particle Physics, 3rd edition. The problem: Consider a tight-binding solid which has alternating atoms of type A and B. The ...
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74 views

Physical Meaning of Over- / Underdetermined Acoustic Eigenvalue Problem

I am performing an eigenmode study on a system of acoustic ducts. The system consists of two large cylindrical volumes connected by several smaller cylindrical volumes (modeling a combustion chamber ...
2
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1answer
45 views

Proton spin independent fine structure “Hamiltonian” $W_f$

To find the perturbation correction (fine structure) in the case of a degenerate energy $E_n^0$, we can diagonalize the operator $W_f^n$, the restriction of $W_f$ to the eigen-space associated to $E_n^...
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145 views

Making An Energy Momentum Plot For A Rashba Model (Using Discretization)

I want to make a plot of the Energy versus the Momentum of the Rashba model, using discrete matrices. First Ill show how I did this for the free particle. Subsequently I will show what goes wrong for ...
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171 views

Adiabatic quantum evolution of single photon or biphoton system

The prerequisite for adiabatic quantum evolution of single photon or biphoton system is as follows. We have to prepare a single photon or biphoton quantum system which has a ground and a higher level ...
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53 views

How to get from angular velocity to acquired phase for neutrino oscillations in matter?

I am reading Akhmedovs 2000 paper on parametric resonance, and I cannot figure out the math of this particular passage: The difference of the neutrino eigenenergies in a matter of density $N_i$ is $...
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69 views

How to solve these coupled differential equations?

I am trying to solve for wavefunctions of 2D tilted Dirac systems, the Hamiltonian for which is: $$\hat H = v_{x}\sigma_{x}\hat p_{x}+v_{y}\sigma_{y}\hat p_{y}+I_{2}(v_{t}^{x}\hat p_{x}+v_{t}^{y}\hat ...
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49 views

Is it possible to derive $2\times 2$ Lorentz transformation matrix from only eigenvectors?

As a preface, I am somewhat familiar with year 1 linear algebra but not too familiar with how one makes the connection to Lorentz transformation matrices so I apologize if the answer is obvious. One ...
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124 views

Numerical exact diagonalization of tight binding Hamiltonian

I want to exactly diagonalize the following Hamiltonian for $10$ number of sites and $4$ number of spinless fermions $$H = -t\sum_i^{L-1} \big[c_i^\dagger c_{i+1} - c_i c_{i+1}^\dagger\big] + V\sum_i^{...
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45 views

Some questions on coherent states and corresponding Hilbert spaces. Reproducing kernal

I have a few questions related to coherent states. I use this source https://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_5.pdf. Using standart inner product $\langle\cdot|\cdot\rangle$ ...
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70 views

Woods-Saxon energy levels

I'm trying to calculate the bound state energy levels for a Woods-Saxon potential, given by: $$V(r) = -\frac{V_{0}}{1+\exp(\frac{r-R}{a})}$$ I'm using Numerov's matrix method, which led me to: $$H\...
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51 views

mutual coherence

I read Vertex-Frequency Analysis on Graphs (arxiv link) Shuman, David I, et al. “Vertex-Frequency Analysis on Graphs.” Applied and Computational Harmonic Analysis, vol. 40, no. 2, 2016, pp. 260–291....
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1answer
108 views

Reconcile a pair of two-qubit boundary-state separability probability analyses

It is now clearly well-established--though formalized proofs are still largely lacking—that the probability, with respect to Hilbert-Schmidt measure, that a generic two-qubit state is separable/...
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39 views

Eigenvalue equation for interacting Green's functions

Studying the articles "Topological Hamiltonian as an exact tool for topological invariants" (https://arxiv.org/abs/1207.7341) and "Simplified Topological Invariants for Interacting Insulators" (https:/...
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76 views

Has the asymptotic theory of eigenvalues of infinite matrixes already been applied to vibrations analysis?

My question is reffering to the masses/springs model of a material, like the one presented in this article http://www.laserpablo.com/baseball/Kagan/UnderstandingCOR-v2.pdf. If one treates a long ...
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112 views

Waveguide eigenmode weak-form (Comsol)

Maxwell equations are: $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t},\,\,\,\,\,\,\,\,\nabla \times \mathbf{H} = +\frac{\partial \mathbf{D}}{\partial t}$$ Subject to the ...
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123 views

Quantum field theory without spectral analysis as first principles

Many systems in physics are studied by looking at a "spectral-abstraction" (i.e. use of a Fourier transform and/or eigenvectors) of some underlying first principles. A good example is a guitar string'...
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149 views

How to estimate the ground state of a potential well when a confinement dimension is added

I have a finite harmonic potential where I trap an electron. The confinement length changes in size. Now, I'm interested in the ground state energy, so I have this 1D Poisson solver which gives me the ...
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80 views

Eigenvalues of Hamiltonian with on-diagonal coordinate

A bit abstract, but if I take the standard graphene Hamiltonian (around the Dirac point) and introduce an on-diagonal term proportional to the coordinate $\hat{y}$, how would I find the eigenstates ...
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1answer
102 views

“Independent simultaneous eigenbras” in Dirac's book 'Principles of Quantum Mechanics'

I've been puzzling through this book off and on and can usually work out what is going on via other external references on the Intertubes. But, this paragraph from pages 55 and 56 has me a bit ...
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167 views

Selecting physical solutions in numerical eigenvalue problems

I try to solve a certain time-independent Schrodinger equation numerically, using the method of finite differences. My boundary conditions are such that the finite difference method gives me an ...
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683 views

Simultaneous eigenket

J. J. Sakurai states in his "Modern Quantum Mechanics", this fact as a theorem ($\pi$ is the parity operator): Suppose $$[H,\pi]=0$$ and $| n>$ is a nondegenerate eigenket of $H$ with ...
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267 views

Eigenstates of operators on constituent systems in tensor product space

Suppose I have two entangled physical systems $\mathcal{A}$ and $\mathcal{B}$ with respective hilbert spaces $\mathcal{H}_{\mathcal{A}}$ and $\mathcal{H}_{\mathcal{B}}$. If $A,B$ are operators on $\...
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57 views

Can an axisymmetric solution produce antisymmetric eigenfunctions?

I'm solving a vibrating membrane. In order to simplify my calculations, it's tempting to assume axisymmetric behaviour. If I solve an axisymmetric problem, am I going to lose all the antisymmetric ...
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429 views

2D quantum well energy spectrum (analytical vs numerical)

I am trying to understand the energy spectrum difference between the analytical and the approximated solution for a quantum well. The particle is inside a box with domain $\Omega=(0,0)$X$(1,1)$. For ...
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28 views

could we obtain the potential (in one dimension) from the Gutzwiller trace?

to solve and obtain the potential of a 1-D Hamiltonian we must solve an integral equation $$ N(E)= A \int_{0}^{E}\frac{V^{-1}(x)}{\sqrt{E-x}}$$ fro a some constant 'A' , then my question is since ...
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62 views

Random quantum systems with asymmetric Lifshitz tails?

For a quantum mechanical system with a periodic Hamiltonian (Schrödinger operator) $H$, let $N(E)$ be its integrated density of states, i.e. the fraction of eigenvalues in the spectrum $\sigma(H)$ ...
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1answer
38 views

Eigenfunctions in Spherically Symmetric Well

I am looking at a problem that has a potential $$ V(r) = \begin{cases} 0 & a<r<b\\ \infty & \text{elsewhere} \end{cases} $$ This is a modification of the infinite spherical well ...
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Infinite Coupled Masses, symmetry, and the simultaneous diagonal theorem for infinite dimensional vector spaces

In The Physics of Waves by Georgi, in Chapter 4, we show that, in a coupled system of masses connected by springs, a transformation that preserves some symmetry $S$ commutes with $K^{-1}M$. From my ...
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34 views

Neumann boundary condition in spherical coordinates

I'm trying to solve heat equation $$\nabla^2 u = \frac{1}{k}\frac{\partial u}{\partial t}$$ in the region $$ a \leq r \leq b, \ \ \ \ 0 \leq \varphi \leq 2\pi, \ \ \ \ 0 \leq \theta \leq \theta_0 $$ ...
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11 views

For vibrations in continuous beam, what is the unit of eigenvalue?

I have been solving a fourth order euler bernoulli differential equation to solve for vibrations of a continuous cantilever beam. When I verified for them using Comsol eigenvalue solver, it gives me ...
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50 views

Eigenvalues of Unitary Matrices

I am considering the standard equation for a unitary transformation $\alpha^* = U \alpha U^{-1}$, where $\alpha$ is an arbitrary linear operator and $U$ is a unitary matrix. Since in quantum ...
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1answer
44 views

Confusion on kinetic energy quadratic forms and eigenfrequencies

I am new to the idea of expressing kinetic energy in terms of the quadratic form. I noticed that online, people often express the kinetic energy as: $$T = \frac{1}{2} \dot q^T M \dot q \tag{1}$$ ...
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11 views

How can I meaningfully diagonalize the eigenvector subspace of a degenerate phonon mode?

It often occurs to find phonon modes which are degenerate by symmetry. In such occasions the eigenvector is usually not physically insightful, as is is a linear combination of the n degenerate ...
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52 views

Applying Sylvester's theorem in quantum mechanics

A $2d$ system consists of $N$ identical cells arranged linearly in series. The transfer matrix of a single cell is an unitary Hermitian $2$x$2$ matrix with eigenvalues $\exp(±i\theta)$. I need to use ...
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29 views

Quantum Potential - how to find its eigenstates?

I am studying the Pöschl-Teller potential in spherical coordinates and doing a change of variable is enough to put it in another sort of differential equation and thus, obtain the solution. The point ...
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51 views

Why is this function an eigenfunction of $\hat{L}_{z}$?

$$\Psi(\varphi)=\frac{1}{\sqrt{2\pi}}(\sin\varphi-\cos\varphi)$$ I am not able to see why the above function is an eigenfunction of $\hat{L}_{z}$ and which is its eigenvalue. I've been trying with ...
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29 views

Applications of Hamiltonians proportional to nth order momentum

Has anyone come across any physical applications of a linear theory where Hamiltonians are proportional to an arbitrary order of momentum? This paper https://michaelberryphysics.files.wordpress.com/...